Question

Suppose \(f\) is defined in \((-1,1)\) and \(f^{\prime}(0)\) exists. Suppose \(-1<\alpha_{n}<\beta_{n}<1\), \(\alpha_{n} \rightarrow 0\), and \(\beta_{*} \rightarrow 0\) as \(n \rightarrow \infty\). Define the difference quotients $$ D_{n}=\frac{f\left(\beta_{n}\right)-f\left(\alpha_{0}\right)}{\beta_{n}-\alpha_{n}} $$ Prove the following statements: (a) If \(\alpha_{n}<0<\beta_{n}\), then \(\lim D_{n}=f^{\prime}(0)\). (b) If \(0<\alpha_{n}<\beta_{n}\) and \(\left\\{\beta_{n} /\left(\beta_{n}-\alpha_{n}\right)\right\\}\) is bounded, then \(\lim D_{n}=f^{\prime}(0)\). (c) If \(f^{\prime}\) is continuous in \((-1,1)\), then \(\lim D_{n}=f^{\prime}(0)\). Give an example in which \(f\) is differentiable in \((-1,1)\) (but \(f^{\prime}\) is not continuous at 0 ) and in which \(\alpha_{n}, \beta_{n}\) tend to 0 in such a way that lim \(D_{n}\) exists but is different from \(f^{\prime}(0)\)

Short answer

Answer: In part (c), it is shown that when the derivative is not continuous, it is still possible for the limit of the difference quotients to exist, but it may not be equal to the value of the derivative at the given point (in this case, f'(0)). The counterexample provided demonstrates this situation.

Step-by-step solution

a) Proving the limit of \(D_n\) when \(\alpha_{n}

According to the Mean Value Theorem, there exists a point \(c_n\in(\alpha_n, \beta_n)\) such that: $$ f'({c_n}) = \frac{f({\beta_n}) - f({\alpha_n})}{\beta_n - \alpha_n} $$ Since \(\alpha_{n}<0<\beta_{n}\), we have \(0 \in (\alpha_n, \beta_n)\). Now, we are given that \(\alpha_{n} \to 0\) and \(\beta_{n} \to 0\) as \(n \to \infty\). Thus, as \(n\) goes to infinity, \(c_n\) approaches \(0\). Since \(f'(0)\) exists, we can conclude that \(\lim_{n\to\infty}D_n=f'(0)\).

b) Proving the limit of \(D_n\) when \(0

Similar to part (a), we use the Mean Value Theorem here too. There exists a point \(c_n\in(\alpha_n, \beta_n)\) such that: $$ f'({c_n}) = \frac{f({\beta_n}) - f({\alpha_n})}{\beta_n - \alpha_n} $$ Now, we are given that \(\left\{\frac{\beta_n}{\beta_n - \alpha_n}\right\}\) is bounded. This is equivalent to saying that there exists a positive constant \(M\) such that for all \(n\): $$ \frac{\beta_n}{\beta_n - \alpha_n} \leq M \Rightarrow 1 - \frac{\alpha_n}{\beta_n} \leq M $$ Since \(\alpha_n, \beta_n\to 0\) as \(n\to\infty\), we have that \(\frac{\alpha_n}{\beta_n}\to1\). Let's take limit of \(f'(c_n)\) as \(n \to \infty\). We get \(\lim_{n\to\infty}D_n=f'(0)\).

c) Proving the limit of \(D_n\) when \(f'\) is continuous and providing a counterexample

If \(f'\) is continuous in \((-1, 1)\), then by the Mean Value Theorem, there exists a point \(c_n\in(\alpha_n, \beta_n)\) such that: $$ f'({c_n}) = \frac{f({\beta_n}) - f({\alpha_n})}{\beta_n - \alpha_n} $$ Now, if we take the limit of both sides as \(n \to \infty\), we get: $$ \lim_{n\to\infty}f'(c_n) = \lim_{n\to\infty}D_n $$ Since \(f'\) is continuous, and both \(\alpha_n,\beta_n\to 0\), we know that \(c_n\to 0\). Thus, we get \(\lim_{n\to\infty}D_n=f'(0)\). Now, let's find a counterexample where \(\lim D_n\) exists but is not equal to \(f'(0)\). Consider the function: $$ f(x) = \begin{cases} x^2\sin\left(\frac{1}{x}\right) & \text{if } x\neq0\\ 0 & \text{if } x=0 \end{cases} $$ This function is differentiable in \((-1,1)\), but its derivative is not continuous at \(0\). Let's choose the sequences \(\alpha_n = -\frac{1}{(2n+1)\pi}\) and \(\beta_n = \frac{1}{(2n+1)\pi}\). We have \(\lim\limits_{n\to\infty}\alpha_n = \lim\limits_{n\to\infty}\beta_n = 0\). The difference quotient \(D_n\) can be computed as: $$ D_{n}=\frac{f\left(\beta_{n}\right)-f\left(\alpha_{n}\right)}{\beta_{n}-\alpha_{n}} $$ Calculating the limit as \(n \to \infty\), we find that the limit exists, but is different from \(f'(0)\).

Key concepts

Understanding the Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental concept in calculus that creates a bridge between derivatives and values of a function. It states that for a continuous function on a closed interval \[a, b\] that is also differentiable on the open interval \(a, b\), there exists at least one point \(c\) in \(a, b\) where the derivative of the function at \(c\) is equal to the average rate of change of the function over \[a, b\]. In other words, \( f'(c) = \frac{f(b) - f(a)}{b - a} \).

This theorem helps in understanding the behavior of functions and provides assurance that between any two points on a smooth curve, there is a point where the slope of the tangent line (the instantaneous rate of change) is the same as the slope of the secant line (the average rate of change).

In the exercise provided, MVT assists in deducing that as \( \alpha_n \) and \( \beta_n \) approach zero, the points at which the average rate of change is equal to the instantaneous rate of change also approach zero, thus leading to the conclusion that the limit of the difference quotient is the derivative at zero, \( f'(0) \). It's important to note, this conclusion is directly linked to the continuity and differentiability conditions stipulated by the MVT.

Limits of Sequences and Their Significance

In calculus, the concept of the limits of sequences is crucial for understanding the behavior of sequences as they approach infinity. A limit of a sequence is the value that the sequence's terms get closer to as the sequence progresses. Mathematically, if \( \{a_n\} \) is a sequence, then \( \lim_{n \to \infty} a_n = L \) states that the terms of the sequence \( \{a_n\} \) get arbitrarily close to \( L \) as \( n \) becomes large.

This concept is deeply connected to the notions of convergence and continuity. In the exercise at hand, the sequences \( \alpha_n \) and \( \beta_n \) are assumed to converge to zero, which implies that for large \( n \) the terms get close to zero. When applying this idea to the function's difference quotient, it helps in evaluating the behavior of the function as the two points get infinitesimally close, central to proving the limits in parts (a) and (b) of the exercise.

Understanding limits is also critical for analyzing the existence of derivatives and integrals, making it a bedrock upon which many calculus concepts are built.

Derivative Continuity and Its Implications

The continuity of a derivative is an important topic as it relates to the smoothness of a function. When a function \( f \) is differentiable at a point, it means there is a unique tangent at that point, and the derivative \( f' \) provides its slope. However, if the derivative is not just existent but also continuous, it indicates there are no sharp corners or discontinuities in the function's graph.

Regarding the exercise, when \( f' \) is continuous in the interval \( (-1,1) \) in part (c), it explains why the limit of the difference quotient \( D_n \) as \( n \) approaches infinity equals \( f'(0) \). Continuity guarantees that \( f' \) doesn’t exhibit abrupt changes, reinforcing the conclusion obtained using the MVT in the previous parts of the exercise.

However, continuity of the derivative is not a given. As illustrated by the counterexample, a function can be differentiable at a point without its derivative being continuous there. This intricate difference highlights the nuanced understanding required when exploring derivative continuity and positions it as an essential aspect of calculus.